Interval of Convergence Calculator
Compute the radius and interval of convergence from ratio or root test limits.
Table of Contents
How to Use
- Enter the series center a
- Provide the ratio/root test limit L
- Choose the test type (ratio or root)
- Calculate to see the radius and open interval; test endpoints separately
Using ratio and root tests
For power series Σ c_n (x - a)^n, the ratio or root test limit L gives the radius of convergence R = 1 / L. If L = 0, the series converges for all x.
- Ratio test: L = lim |c_{n+1} / c_n|
- Root test: L = lim |c_n|^{1/n}
- Radius: R = 1 / L (if L ≠ 0)
Remember to test endpoints
The interval of convergence is typically (a - R, a + R). Convergence at x = a ± R depends on separate tests, such as alternating series, p-series, or comparison tests.
Document which endpoints converge to describe the full closed or half-open interval.
Frequently Asked Questions
- What if L = 0?
- L = 0 means terms shrink faster than any geometric sequence, so the radius is infinite and the series converges for all x.
- What if the limit does not exist?
- The ratio/root tests require a limit. If it oscillates or diverges, you may need a different test or to analyze subsequences.
- How do I handle endpoints?
- Plug x = a ± R into the series and test separately. The result may converge at neither, one, or both endpoints, changing the final interval notation.